A052344 - OEIS

A052344

Number of ways to write n as the unordered sum of two nonzero triangular numbers.

0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 2, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 2, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 2, 1, 0, 0, 2, 0, 1, 1, 0, 2, 0, 0, 0, 1, 2, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 2, 1, 0, 0, 2, 0, 0, 1, 0, 3, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 2, 0, 0, 1, 0, 1, 1, 1

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OFFSET

0,17

COMMENTS

Number of ways to write 8*n+2 as the unordered sum of two odd squares > 1. - Robert Israel, Feb 24 2016

Number of partitions of 2n into two promic numbers > 1. - Wesley Ivan Hurt, Jun 09 2021

LINKS

T. D. Noe, Table of n, a(n) for n = 0..10000

FORMULA

G.f.: (Theta_2(sqrt(x))^2 - 4*x^(1/8)*Theta_2(sqrt(x)) + 2*Theta_2(x))/(8*x^(1/4)) where Theta_2 is a Jacobi theta function. - Robert Israel, Feb 24 2016

a(n) = Sum_{k=1..n} c(k) * c(2*n-k), where c(n) is the characteristic function of promic numbers (A005369). - Wesley Ivan Hurt, Jun 09 2021

a(n) = Sum_{k=1..floor(n/2)} c(k) * c(n-k), where c = A010054. - Wesley Ivan Hurt, Jan 06 2024

MAPLE

G:= (1/8)*(JacobiTheta2(0, sqrt(q))^2-4*JacobiTheta2(0, sqrt(q))*q^(1/8)+2*JacobiTheta2(0, q))/q^(1/4):

S:= series(G, q, 1001):

seq(coeff(S, q, j), j=0..1000); # Robert Israel, Feb 24 2016

MATHEMATICA

nn=150; tri=Accumulate[Range[nn]]; t=Table[0, {tri[[-1]]}]; Do[n=tri[[i]]+tri[[j]]; If[n <= tri[[-1]], t[[n]]++], {i, nn}, {j, i}]; t=Prepend[t, 0]

CROSSREFS

Cf. A000217, A052343-A052348, A053587, A056303, A056304.

Column k=2 of A319797.

Cf. A005369, A010054.

Sequence in context: A317748 A090465 A364357 * A339375 A147768 A167746

Adjacent sequences: A052341 A052342 A052343 * A052345 A052346 A052347

KEYWORD

nonn

AUTHOR

Christian G. Bower, Jan 23 2000

STATUS

approved